As is known in the art, computationally secure cryptosystems, which are largely based upon unproven hardness assumptions, have led to cryptographic schemes that are widely adopted and thrive from both a theoretical and a practical perspective in communication systems. Such cryptographic schemes are used millions of times per day in applications ranging from online banking transactions to digital rights management. Increasing demands for large-scale high-speed data communications, for example, have made it important for communication systems to achieve efficient, reliable, and secure data transmissions.
As is also known, information-theoretic approaches to secure cryptosystems, particularly secrecy, are traditionally concerned with unconditionally secure systems, i.e. systems with schemes that manage to hide all bits of a message from an eavesdropper with unlimited computational resources available to intercept or decode a given message. It is well known, however, that in a noiseless setting unconditional secrecy (i.e., perfect secrecy) can only be attained when both a transmitting party and a receiving party share a random key with entropy at least as large as the message itself (see, e.g., “Communication Theory of Secrecy Systems,” by C. E. Shannon, Bell Systems Technical Journal, vol. 28, no. 4, pp. 656-715, 1949). It is also well known that, in other cases, unconditional secrecy can be achieved by exploiting particular characteristics of a given scheme, such as when a transmitting party has a less noisy channel (e.g., wiretap channel) than an eavesdropper. (see, e.g., “Information Theoretic Security,” by Liang et al., Found. Trends Commun. Inf. Theory, vol. 5, pp. 355-580, April 2009).
Traditional secrecy schemes, including secure network coding schemes and wiretap models, assume that an eavesdropper has incomplete access to information needed to intercept or decode a given data file. Wiretap channel II, for example, which was introduced by L. Ozarow and A. Wyner, is a wiretap model that assumes an eavesdropper observes a set k out of n transmitted symbols (see, e.g., “Wiretap Channel II,” by Ozarow et al, Advances in Cryptography, 1985, pp. 33-50). Such wiretap model was shown to achieve perfect secrecy, but practical considerations limited its success. An improved version of Wiretap channel II was later developed by N. Cai and R. Yeung, which addressed a related problem of designing an information-theoretically secure linear network code when an eavesdropper can observe a certain number of edges in the network (see, e.g., “Secure Network Coding,” by Cai et al., IEEE International Symposium on Information Theory, 2002).
A similar and more practical approach was later described in “Random Linear Network Coding: A Free Cipher?” by Lima at al. in IEEE International Symposium on Information Theory, June 2007, pp. 546-550. However, with an ever increasing amount of data being streamed over the internet and in both near and far-field communications, for example, there remains a need for new and more efficient methods and systems for use in providing secure communication in communications systems and networks. Additionally, there remains a need for characterizing and optimizing such secrecy schemes through improved information-theoretic metrics.